Distance between points MCQ Quiz - Objective Question with Answer for Distance between points - Download Free PDF

Answer : 3

Solution :

We have, equation of line.

 and plane : x - y + z = 16

any point on line is (3t + 2, 4t - 1, 12t + 2) and this point will satisfy the plane.

∴ 3t + 2 - 4t + 1 + 12t + 2 = 16 ⇒ 11t = 11 ⇒ t = 1

So, point will be (5, 3, 14)

Hence, distance between (5, 3, 14) and (1, 6, 2) is

= 

=  = 13 units

Calculation

Let

Any point on the line can be written in the parametric form as

To find the point of intersection, let us substitute the point in the equation of the plane.

Hence, the point of intersection is

The distance of from

Hence option 4 is correct

Concept:

• Line and Plane Intersection:
  • A line in 3D can be written in symmetric form: (x − x₁)/a = (y − y₁)/b = (z − z₁)/c
  • A plane in 3D has the general form: Ax + By + Cz + D = 0
  • To find the intersection point, substitute parametric equations of the line into the plane's equation.

Calculation:

Given, line: (x + 1) = (y + 3)/3 = (−z + 2)/2

Let the common value = t

⇒ x = t − 1

⇒ y = 3t − 3

⇒ z = 2 − 2t

Given plane: 3x + 4y + 5z = 10

Substitute values of x, y, z into the plane:

⇒ 3(t − 1) + 4(3t − 3) + 5(2 − 2t) = 10

⇒ 3t − 3 + 12t − 12 + 10 − 10t = 10

⇒ (3t + 12t − 10t) + (−3 −12 + 10) = 10

⇒ 5t − 5 = 10

⇒ 5t = 15

⇒ t = 3

Now, find coordinates:

⇒ x = 3 − 1 = 2

⇒ y = 3×3 − 3 = 6

⇒ z = 2 − 2×3 = −4

∴ The point of intersection is (2, 6, −4)

Calculation

Equation of line AB

Distance of P from A is 21

⇒ 

⇒ 

⇒ (22, 0, 7) = (a, b, c) 

The distance between the points P(22, 0, 7) and Q(4, –12, 3) 

 22

Formula Used:

To find the distance between two points (x₁, y₁) and (x₂, y₂) in a Cartesian coordinate system, you can use the distance formula:

Distance (d) = 

Explanation:

In this case, the points are (2, 3) and (4, 1), so you can use these coordinates in the distance formula:

⇒ Distance (d) = 

⇒ Distance (d) = 

⇒ Distance (d) = √(4 + 4)

⇒ Distance (d) = √8

⇒ Distance (d) = 2√2

So, the distance between the points (2, 3) and (4, 1) is 2√2 units. 

CONCEPT:

If A(x1, y1, z1) and B(x2, y2, z2) then the distance between the points A and B is given by: 

CALCULATION:

Given: A (2, 0, 3) and B (- 4, a, - 1) are two points in a 3D space such that distance between them is 8 units.

As we know that, if A(x1, y1, z1) and B(x2, y2, z2) then the distance between the points A and B is given by: 

CONCEPT:

If A(x1, y1, z1) and B(x2, y2, z2) then the distance between the points A and B is given by: 

CALCULATION:

Given: P (6, 4, - 3) and Q (2, - 8, 3) are two points in a 3D space.

Here, we have to find the distance between the the given points.

As we know that, if A(x1, y1, z1) and B(x2, y2, z2) then the distance between the points A and B is given by: 

⇒ 

Concept:

Distance formula between the points (x1, y1, z1) and (x2, y2, z2) is = 

Explanation:

We have to find the distance between the point P (2m, 3m, 4m) and the x-axis.

Now,

The General point in the x-axis can be represented as (x, 0, 0) (As the point is in x-axis, its y and z

coordinates will be 0)

⇒ We have to find the distance between the point P (2m, 3m, 4m) and (2m, 0, 0).

Thus,

Distance between the point P (2m, 3m, 4m) and the x-axis =

Distance between the point P (2m, 3m, 4m) and the (2m, 0, 0) =

⇒ 5m

CONCEPT:

The distance between the points A(x1, y1, z1) and B(x2, y2, z2) is given by:

CALCULATION:

Let P(x, y, z) be the point which is equidistant from the points A(3, 4, -5) and B(-2, 1, 4)

As we know that, the distance between the points A(x1, y1, z1) and B(x2, y2, z2) is given by:

First let's find out the distance between the the points P and A

⇒ 

Similarly, let's find out the distance between the the points P and B

⇒ 

∵ PA = PB

⇒ 

By squaring both the sides we get,

⇒ (3 - x)2 + (4 - y)2 + (-5 - z)2 = (-2 - x)2 + (1 - y)2 + (4 - z)2

⇒ x2 + y2 + z2 - 6x - 8y + 10z + 50 = x2 + y2 + z2 + 4x - 2y - 8z + 21

⇒ 10x + 6y - 18z - 29 = 0

So, the set of the points which are equidistant from the points A(3, 4, -5) and B(-2, 1, 4) is given by: 10x + 6y - 18z - 29 = 0.

Hence, correct option is 2.

CONCEPT:

• The distance between the points A(x1, y1, z1) and B(x2, y2, z2) is given by: 

CALCULATION:

Given: A(2, - 1, 3) and B(-2, 1, 3) are two points in a 3D plane.

Let d denote the distance between the given points.

As we know that, the distance between the points A(x1, y1, z1) and B(x2, y2, z2) is given by: 

Here, x1 = 2, y1 = - 1, z1 = 3, x2 = - 2, y2 = 1 and z2 = 3.

⇒ 

⇒ 

Hence, correct option is 1.

CONCEPT:

• The distance between the points A(x1, y1, z1) and B(x2, y2, z2) is given by: 

CALCULATION:

Given: The distance between the points A(- 1, 3, - 4) and B(1, - 3, a) is 

Let d denote the distance between the given points.

As we know that, the distance between the points A(x1, y1, z1) and B(x2, y2, z2) is given by: 

Here, x1 = -1, y1 = 3, z1 = -4, x2 = 1, y2 = - 3, z2 = a and 

⇒ 

⇒ 

By squaring both the sides we get,

⇒ 104 = 40 + a² + 8a + 16

⇒ a² + 8a - 48 = 0

⇒ a = 4, - 12

Hence, correct option is 2.

CONCEPT:

If A(x1, y1, z1) and B(x2, y2, z2) then the distance between the points A and B is given by: 

CALCULATION:

Given: P (2, - 5, 7) and Q (3, 4, 5) are two points in a 3D space.

Here, we have to find the distance between the the given points.

As we know that, if A(x1, y1, z1) and B(x2, y2, z2) then the distance between the points A and B is given by: 

Given:

Point 1 = (2, -1, 3)

Point 2 = (-5, 2, 1)

Formula:

Distance between two points having coordinates (x₁, y₁, z₁) and (x₂, y₂ , z₂) is given by :

Solution:

d = √[(2 - (-5))² + ((-1) - 2)² + (3 - 1)²]

= √(49 + 9 + 4)

= √62 units

Note: The options given in the Official Paper were wrong. We have corrected the option.

CONCEPT:

If A(x1, y1, z1) and B(x2, y2, z2) then the distance between the points A and B is given by: 

CALCULATION:

Given: A (1, 2, 5) and B (3, - 5, 0) are two points in a 3D space.

Here, we have to find the distance between the the given points.

As we know that, if A(x1, y1, z1) and B(x2, y2, z2) then the distance between the points A and B is given by: 

Given:

Two points are (0, 5, 5) and (5, 8, 6).

Concept:

Distance between two points  is

Calculation:

Distance form origin (0,0,0) to point (0,5,5) is

Distance form origin (0,0,0) to point (5,8,6) is

Now the sum of distances from origin to (0, 5, 5) and (5, 8, 6) is

Hence the option (4) is correct.